Logic
— the Art of Reasoning

Mathematics
— the Art of Studying Patterns Using Logic

Finding the Greatest Common
Factor Visually, without Calculation

Common methods for
finding the Greatest Common Factor (GCF), a.k.a. Greatest Common Divisor
(GCD), employ factording. Factoring is a process that requires repeated
integer division having integer results. First you divide the source number
by any integer you see fit. Then you proceed to divide the dividend and/or
the divisor. You repeat this orcess until none of the divisor and dividends
can be divided any more. [A process that is applied repeatedly to the results
of its previous itterations, such as this one, is called recursive.]

The alternative
method described here is graphic. It requires no division. In fact, as
long as you draw to scale, this visual method requires no computation whatsoever.

This graphic method
makes use of the relationship between geometry and arithmatic.

This method has several
advantages:

Using graphics, this method
provides a visual alternative for computational methods.

It also provides a visual
means to explain the concept of greatest common factor/divisor (GCF/GCD)
and why the various methods, including those that employ factoring, work.

Being an alternative,
it demonstrates that there are different ways to reach the same mathematical
destination.

This method demonstrates
the close relationship between arithmatic and geometry.

This method is simpler
than those requiring factoring because no division is done.

More Details

When given two integers,
m
and
n,
the common methods to find their GCF (or GCD) is to divide both numbers
by the most obvious factors (excluding 1)
recursively until all are exhausted. Then GCF is equal to the product of
all of the common factors. For example, say m
and
n
are 36
and 60 respectively.
Divide both numbers by 2.
The results, 18
and 30,
are also even so divide them by 2
again. This time you get 9
and 15,
both of which are divisible by 3.
If other obvious factors are known, such as 5,
9,
7 or 11,
then divide by these factors. The problem is that often no factor is easily
apparent for both m
and n, especially
when at least one of the numbers is larger than 12x12,
which is the top of "math facts" with which students are required to fluent.This
method is
visual -- with it you find the greatest common
factor without doing any calculation — you only need to know and
to do is to draw to scale. Using graph paper or other means for
greating a grid (e.g., snap cubes) you can still use this method; in this
case all you need to know and do is count. You draw and the solution
appears. When teaching visually-oriented students, who are experiencing
difficulties understanding the computational method, employing this graphic
method, is often easier.

Example
1: What is the greatest common factor of 12
and 20?

Example 2: What is
the greatest common factor of 7
and 25?

Answer: 7
and 25
do not have a common factor.(Remember, since 1
is a factor of every number, it is not helpful for applications that require
common factors and, therefore, we do not include it in this case.)

The General Procedure:

Say the two numbers
for which you want to find the GCF are m
and n.
Let us say that m
> n. (the
case for m
< n
is symetric and we can reverse the roles of the numbers either in the inequality
or in the procedure. If m
= n
then there is nothing to do because the LCF = m
= n.)

Draw an m
by n
rectangle;

Draw the largest possible
square inside this rectangle -- that is, draw a nxn
square -- such that one of its corners coincides with one of the corners
of the rectangle. We call the leftover rectangle the remaining rectangle.
In this case the dimensions of the remaining rectangle are nx(m-n),
where
n
>= (m-n).

If the remaining rectangle
is a square -- that is n = (m-n)
-- stop. Your GCF equals
the length of one edge of this square, which I call the GCF square.

Otherwise, repeat Step
2 and Step 3 in each remaining rectangle. In other words, as long as n
> (m-n), substitute n
for m
and (m-n) for
n
and
repeat Step 2.

To confirm the result,
replicate the GCF square along each edge of the large rectangle, starting
at one corner. The GCF square should fit along each side
an integer number of times. Putting it in mathematical terms,
if the GCF square fits along the m
edge p
times and along the n
edge q
times, then both p
and q
must be integers. If even one of them is not, then there is a mistake in
the drawing and either the result or the confirmation is incorrect.

To Observe and
Figure-out Why

Notice that p
and q
are not equal.

Also, most likely neither
p
nor q
is equal to 1.
If either equals 1 then
the longer edge is a multiple of the short edge.

Why This Works?

Explanation 1.
By replicating The GCF square along the edges of the original rectangle
during the confirmation phase (Step 5) you perform division — you divide
the length of the edge by the size of the GCF square. This step clearly
shows that the GCF square is indeed a common factor (divisor). It is the
largest
common
factor (divisor) because during the procedure you tested every possible
number that is a factor (divisor) of one of the rectangle edges by representing
these numbers with squares of diminishing sizes.

Explanation 2. When
you repeatedly draw the same largest possible square in a rectangle
(this happens once in Example 1 with the 4x4
squares and twice in Example 2 with the 7x7
and with the 1x1
squares), you perform division by doing repeated subtraction.
That is, this method substitutes repeated subtraction for division. As
such, it is simpler then having to divide. Furthermore, since no division
is required, you also do not have to figure out any divisor (factor) of
either number. It is a simplified version of the procedure known as Euclidean
algorithm.

Finding the LCF (LCD)
of Large Numbers?

Large numbers pose
a practical limitation for this method. Drawing to scale, factoring large
numbers with this method requires a correspondingly large drawing surface
(e.g., sheet of paper, whiteboard) and/or very precise minute-drawing skill
(using such tools as graph paper, snap cubes, whiteboard and a ruler).
If either such tools or skills are not available, then one is limited with
resepct to the size of the numbers he or she can factor.

There is a way around
this limitation.

Once you understand
and are comfortable with this graphic method, you can draw not to scale
and use it to figure out the GCF of any two integers. The trade off is
that when you factor large number using not-to-scale drawing, you must
perform subtractions, each time you draw a "not to scale square", you subtract
its size from the the size of the rectangle's edge along which you draw
it. In a sense, what you have to do is explicitly compute the subtraction
that, when drawing to scale, the act of drawing performs for you implicitly.

Although you must compute,
subraction is simpler than division. Moreover subraction is straight-forward
comparing with figuring out the divisors (factors) of large numbers, which
for most people is a time-consuming and tedious trail-and-error process.

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